I was wondering which one is more general, metric linear spaces or locally convex topological vector spaces?
Is a metric linear space a locally convex topological vector space? Vice versa?
In terms of the number of books and websites that have mentioning, it seems like metric linear spaces are less popular than locally convex topological vector spaces, although both are topological vector spaces and normed vector spaces are both, doesn't it? Why is that?
For example, there is no Wiki article for metric linear spaces, but there is one for locally convex topological vector spaces.
- As to metric linear spaces, the definition I saw from this book says a metric linear space is a vector space with a metric, such that addition and scalar multiplication are both continuous wrt the metric. That the metric is translation invariant is stronger than that definition. Isn't it?
Thanks and regards!